Continuum Mechanics and Theory of Materials pp 413-452 | Cite as

# Plasticity

Chapter

## Abstract

The theory of *plasticity* ^{1} models rate-independent material behaviour with hysteresis. Plasticity expands on the theory of elasticity by taking internal dissipation into consideration. However, internal friction is represented from a completely different viewpoint here, compared with viscoelasticity: whereas viscoelastic material behaviour is characterised by its fading memory properties, the theory of plasticity tends to express a perfect memory of a material body.

## Keywords

Evolution Equation Strain Amplitude Yield Surface Internal Variable Flow Rule
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## References

- 1.See Green & Naghdi [1965, 1971]. A comprehensive presentation of the theory of plasticity is given by Lubliner [1990]. See also Khan & Huang [1995] and Levitas [1993]. The development of the plasticity theory is reviewed critically in Naghdi [1990], where many references are to be found. Bertram [1999] develops a fairly general concept of elastoplasticity, discussing the extent to which the current state of finite plasticity fits into the scope of a constitutive theory within the tradition of rational mechanics.Google Scholar
- 2.The same statement was expounded along the same lines by Pipkin & Rivlin [1965]. See also Owen & Williams [1968].Google Scholar
- 3.The arclength representation was applied extensively by K.C. Valanis. In particular, he introduced a material-dependent arclength for the purpose of describing rate-independent hysteresis and nonlinear hardening behaviour under mechanical and thermomechanical points of view. See Valanis [1971,1974,1975,1977,1980,1981,1995].Google Scholar
- 4.Cf. Valanis [1995], eq. (63).Google Scholar
- 5.A similar analysis can be found in Valanis [1971], pp. 542.Google Scholar
- 6.A representation by means of a generalised arclength — a constitutive function of the kinematic arclength — was proposed by Valanis [1971] under the heading
*Endochronic Theory of Plasticity.*See Valanis [1971, 1975, 1977, 1980], Rivlin [1981]; Valanis [1981]; Haupt [1977].Google Scholar - 7.Experimental evidence of cyclic hardening and appropriate constitutive modelling is presented in Bruhns et al. [1992].Google Scholar
- 8.Evolution equations of this kind were developed by Kamlah [1994]. See Haupt et al. [1991, 1992a]; Haupt & Kamlah [1995]. The concept of a generalised arclength was successfully applied by Lion [1994]; Lührs [1997], pp. 24; Lührs & Haupt [1997] and Hartmann et al. [1998].Google Scholar
- 9.
- 10.Strain space formulations of finite elastoplasticity were developed and successfully applied by Besdo [1981,1989].Google Scholar
- 11.See, for example, Haupt & Kamlah [1995], pp. 285.Google Scholar
- 12.See Ehlers [1993], pp. 389.Google Scholar
- 13.The evolution equation (11.58) for the generalised arclength is constructed so as to satisfy the consistency condition.Google Scholar
- 14.See, for example, Lubliner [1990], pp. 125; Ehlers [1993], pp. 366. More recent investigations are reported in Streilein [1997].Google Scholar
- 15.The state of the art in modelling finite elastoplasticity is reviewed in Naghdi [1990]. See also the papers Simo [1988]; Miehe & Stein [1992]; Miehe [1993] Le & Stumpf [1993]. 16 The idea for this decomposition originates from Kröner [1960]. See also Lee [1969].Google Scholar
- 17.The somewhat sophisticated interpretation of the multiplicative decomposition is replaced by the more abstract assumption of a
*material isomorphism*in the work of Bertram [1999]. See also Svendson [1998].Google Scholar - 18.In this connection, anisotropic elasticity properties would have to be represented by isotropic tensor functions depending not only on the elastic strain but on other variables, known as structural tensors, besides. See Horz [1994], pp. 42; Horz et al. [1994]. Ways in which anisotropic elasticity may be taken into consideration without recourse to structural tensors are discussed in Section 13. 6.Google Scholar
- 19.Ehlers [1993], pp. 389.Google Scholar
- 20.Constitutive models of this structure have been applied in the context of analytical investigations by Bonn [1992] and in connection with numerical solutions of boundary value problems by Hartmann [1993].Google Scholar

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